Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered height ha, height hb and height hc.

Acute scalene triangle.

Sides: a = 264.3933198314   b = 254.2244229148   c = 132.1976599157

Area: T = 16524.57548946
Perimeter: p = 650.8144026618
Semiperimeter: s = 325.4077013309

Angle ∠ A = α = 79.54106163266° = 79°32'26″ = 1.3888245644 rad
Angle ∠ B = β = 71.00875361503° = 71°27″ = 1.23993152996 rad
Angle ∠ C = γ = 29.45218475231° = 29°27'7″ = 0.51440317101 rad

Height: ha = 125
Height: hb = 130
Height: hc = 250

Median: ma = 153.548806716
Median: mb = 165.9298787135
Median: mc = 250.794448916

Inradius: r = 50.781125
Circumradius: R = 134.4330314066

Vertex coordinates: A[132.1976599157; 0] B[0; 0] C[86.04551237116; 250]
Centroid: CG[72.74772409561; 83.33333333333]
Coordinates of the circumscribed circle: U[66.09882995784; 117.0587781172]
Coordinates of the inscribed circle: I[71.18327841614; 50.781125]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 100.4599383673° = 100°27'34″ = 1.3888245644 rad
∠ B' = β' = 108.992246385° = 108°59'33″ = 1.23993152996 rad
∠ C' = γ' = 150.5488152477° = 150°32'53″ = 0.51440317101 rad

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How did we calculate this triangle?

1. Input data entered: height ha, height hb and height hc.

h_a = 125 ; ; h_b = 130 ; ; h_c = 250 ; ;

2. From height ha, height hb and height hc we calculate area T:

h = fraction{ fraction{ 1 }{ h_a } + fraction{ 1 }{ h_b } + fraction{ 1 }{ h_c } }{ 2 } = (1 / 125 + 1 / 130 + 1 / 250) / 2 = 0.01 ; ; ; ; S_1 = 4 * sqrt{ h * (h - 1 / h_a) * (h - 1 / h_b) * (h - 1 / h_c) } ; ; ; ; S_1 = 4 * sqrt{ 0.01 * (0.01 - 1 / 125) * (0.01 - 1 / 130) * (0.01 - 1 / 250) } = 6.052 * 10**{ -5 } ; ; ; ; T = fraction{ 1 }{ S_1 } = fraction{ 1 }{ 6.052 * 10**{ -5 } } = 16524.575 ; ;

3. From area T and height ha we calculate side a:

T = fraction{ a h_a }{ 2 } ; ; ; ; a = fraction{ 2 T }{ h_a } = fraction{ 2 * 16524.575 }{ 125 } = 264.393 ; ;

4. From area T and height hb we calculate side b:

T = fraction{ b h_b }{ 2 } ; ; ; ; b = fraction{ 2 T }{ h_b } = fraction{ 2 * 16524.575 }{ 130 } = 254.224 ; ;

5. From area T and height hc we calculate side c:

T = fraction{ c h_c }{ 2 } ; ; ; ; c = fraction{ 2 T }{ h_c } = fraction{ 2 * 16524.575 }{ 250 } = 132.197 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 264.39 ; ; b = 254.22 ; ; c = 132.2 ; ;

6. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 264.39+254.22+132.2 = 650.81 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 650.81 }{ 2 } = 325.41 ; ;

8. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 325.41 * (325.41-264.39)(325.41-254.22)(325.41-132.2) } ; ; T = sqrt{ 273061575.45 } = 16524.57 ; ;

9. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16524.57 }{ 264.39 } = 125 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16524.57 }{ 254.22 } = 130 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16524.57 }{ 132.2 } = 250 ; ;

10. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 254.22**2+132.2**2-264.39**2 }{ 2 * 254.22 * 132.2 } ) = 79° 32'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 264.39**2+132.2**2-254.22**2 }{ 2 * 264.39 * 132.2 } ) = 71° 27" ; ; gamma = 180° - alpha - beta = 180° - 79° 32'26" - 71° 27" = 29° 27'7" ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16524.57 }{ 325.41 } = 50.78 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 264.39 }{ 2 * sin 79° 32'26" } = 134.43 ; ;

13. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 254.22**2+2 * 132.2**2 - 264.39**2 } }{ 2 } = 153.548 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 132.2**2+2 * 264.39**2 - 254.22**2 } }{ 2 } = 165.929 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 254.22**2+2 * 264.39**2 - 132.2**2 } }{ 2 } = 250.794 ; ;
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